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Simple machine

A simple machine is a basic mechanical device with few or no moving parts that alters the magnitude or direction of a force to perform work more efficiently, without reducing the total amount of work required.^[1] The six classical types of simple machines are the lever, wheel and axle, pulley, inclined plane, wedge, and screw, each providing mechanical advantage—the ratio of output force to input force—by trading force for distance or changing force direction.^[1] These devices form the foundational building blocks of more complex machinery, enabling humans to amplify physical effort in tasks ranging from lifting heavy objects to cutting materials.^[2] The concept of simple machines traces back to ancient civilizations, particularly ancient Greece, where philosophers and engineers like Archimedes (c. 287–212 BCE) explored principles such as the lever and screw through mathematical treatises, laying the groundwork for classical mechanics.^[3] Archimedes famously demonstrated the lever's potential by claiming, "Give me a place to stand, and I shall move the Earth," highlighting its ability to achieve vast mechanical advantages with precise fulcrum placement.^[4] Over centuries, from the Renaissance onward, engineers viewed all mechanisms as combinations of these six types, influencing fields like engineering, physics, and everyday technology.^[3] In physics, simple machines illustrate key principles of energy conservation and work, where the input work (force times distance) equals the output work, minus any frictional losses.^[1] For instance, an inclined plane reduces the force needed to lift an object by increasing the distance traveled, while a pulley system redirects force to lift loads vertically with less effort.^[1] Their study remains essential in education and engineering, demonstrating how fundamental tools underpin modern innovations like cranes, vehicles, and robotic systems.^[1]

Definition and Types

Core Definition

A simple machine is a basic mechanical device that provides a mechanical advantage by altering the direction or magnitude of an applied force, enabling the performance of work with reduced effort, typically through a trade-off in distance.^[1] These devices operate without complex assemblies, relying on fundamental principles of force transmission to multiply or redirect input forces.^[5] Key characteristics of simple machines include minimal or no moving parts—often just one—allowing them to transmit forces and motions efficiently in straightforward ways.^[6] They serve as foundational elements for more intricate machinery, performing work by applying leverage or redirection rather than generating energy./09%3A_Statics_and_Torque/9.05%3A_Simple_Machines) In ideal cases, simple machines conserve work, meaning the input work equals the output work, where work W is calculated as force F times distance d (W = F \times d).^[1] This conservation principle underscores their role in balancing effort and load without energy loss in theoretical models. The six classical types—lever, wheel and axle, pulley, inclined plane, wedge, and screw—exemplify these principles./09%3A_Statics_and_Torque/9.05%3A_Simple_Machines)

The Six Classical Types

The six classical simple machines, defined by Renaissance scientists, are the lever, wheel and axle, pulley, inclined plane, wedge, and screw; these archetypal devices transform the magnitude or direction of an applied force with minimal components.^[7] These six types were formally defined as the classical simple machines by scientists during the Renaissance period, building on principles explored in ancient Greece.^[7] Each operates on the principle of mechanical advantage, allowing work to be performed more efficiently by trading force for distance or altering force direction.^[5] Attributed in part to the ancient Greek mathematician Archimedes for their foundational analysis, these machines form the basis for more complex systems. Lever. The lever consists of a rigid bar or beam that pivots about a fixed point called the fulcrum, enabling a small input force applied at one end to lift a larger load at the other.^[8] By positioning the fulcrum strategically—such as between the load and effort (first-class lever), at the end (second-class), or with the effort between fulcrum and load (third-class)—it multiplies force through torque balance.^[8] Archimedes is credited with defining the lever's principle in the 3rd century BCE, famously stating, "Give me a place to stand, and I shall move the Earth."^[9] Common examples include seesaws, crowbars, and oars, where illustrations typically depict the bar, fulcrum, effort arm, and load arm to show force application. Wheel and Axle. This machine comprises two rigidly connected rotating cylinders or wheels of different diameters—a large wheel and a smaller central axle—allowing a force applied to the wheel's rim to produce greater torque on the axle.^[10] It facilitates rotational motion, reducing the effort needed to turn or pull loads, as in doorknobs, steering wheels, or windlasses.^[11] Diagrams often illustrate the radius difference, with force vectors showing how the larger radius amplifies torque while the system rotates together. Pulley. A pulley is a wheel mounted on an axis with a grooved rim over which a rope, chain, or belt runs, primarily to redirect the direction of an applied force rather than multiply it significantly in single configurations.^[12] Compound pulleys, involving multiple wheels and ropes, can achieve greater mechanical advantage by distributing the load.^[12] Archimedes is credited with inventing the compound pulley system in the 3rd century BCE, using it for lifting heavy loads like ships.^[9] Examples include flag hoists, elevators, and cranes, where visuals highlight the rope path and tension equality across segments. Inclined Plane. The inclined plane is a flat, sloped surface that reduces the force required to raise an object by spreading the effort over a longer distance along the slope, rather than lifting vertically.^[5] The steeper the angle, the greater the force needed, but the shorter the distance; it trades input work for easier application.^[5] Ramps and sloped roads exemplify this, with diagrams showing parallel force components and height versus base length to clarify load elevation. The wedge and screw derive from the inclined plane, adapting its principles for specialized tasks like separation and conversion of motion types.^[13] Wedge. Formed by two inclined planes joined at their thin ends to create a V-shape, the wedge applies force to split, hold, or secure objects by driving it into a material, where the planes convert sideways effort into upward separation.^[13] It excels in tasks requiring penetration with minimal perpendicular force, such as axes, knives, or chisels.^[11] Illustrations commonly depict the wedge inserted into a block, with arrows indicating input force along the length and resulting splitting action. Screw. The screw functions as an inclined plane coiled around a cylindrical shaft, transforming rotational motion into linear advancement or vice versa through its threaded surface.^[14] Twisting the screw drives it forward with less axial force but over multiple turns, as seen in bolts, jar lids, or Archimedes' screw for water lifting.^[14] Visual aids often unwind the thread to reveal the underlying inclined plane, emphasizing the helix's pitch and rotation direction.

Historical Development

Ancient and Classical Contributions

The ancient Egyptians employed simple machines such as levers and ramps in monumental construction projects, notably during the building of the pyramids around 2600 BCE. For instance, workers used wooden levers to maneuver and position massive stone blocks, while ramps—essentially inclined planes—facilitated the transport of these blocks to higher levels of structures like the Great Pyramid of Giza.^[15]^[16] In ancient Greece during the 4th century BCE, philosophers attributed to Aristotle explored the principles of levers in works like the Mechanical Problems, recognizing their role in providing mechanical advantage through the balance of forces at different distances from the fulcrum. This early analysis laid foundational ideas for understanding how levers could amplify effort, though the text is now considered pseudo-Aristotelian but reflective of contemporary thought.^[17] Archimedes, in the 3rd century BCE, advanced these concepts significantly in his treatise On the Equilibrium of Planes, where he formalized the law of the lever, stating that a balanced beam produces equal torques on either side, thereby quantifying mechanical advantage. He also examined pulleys and the screw, demonstrating their practical applications in lifting and propulsion, which built on and refined earlier Greek insights.^[18]^[19] These developments had profound cultural impacts, enabling engineering feats such as the construction of aqueducts in the classical Roman period, where levers and pulleys aided in assembling stone arches and channels over vast distances. Similarly, simple machines were integral to siege engines, like the compound pulleys and levers Archimedes designed to defend Syracuse against Roman forces in 213 BCE, showcasing their strategic value in warfare.^[20]^[21]

Renaissance to Industrial Era

During the Renaissance, Leonardo da Vinci contributed significantly to the understanding of simple machines through his detailed sketches of mechanical devices, including innovative designs for screws and gear systems that demonstrated practical applications in engineering.^[22]^[23] In the late 15th century, da Vinci's notebooks, such as the Codex Madrid, illustrated how screws could be adapted for lifting and propulsion, while gears enabled efficient force transmission in proto-machines like lathes, laying groundwork for more complex mechanisms.^[24] These visualizations emphasized the integration of simple elements to achieve mechanical advantage, influencing subsequent inventors by showing machines as assemblies of basic components. In the late 16th century, Simon Stevin advanced the theoretical foundation of simple machines with his 1586 treatise De Beghinselen der Waterwichticheyd (Elements of Hydrostatics), where he derived the law of forces acting on inclined planes using a chain of spheres to demonstrate equilibrium.^[25] Stevin's "clootcrans" experiment proved that the force required to balance a weight on an inclined plane is inversely proportional to the plane's length, providing a mathematical basis for calculating mechanical advantage without relying on ancient qualitative descriptions.^[26] This work bridged hydrostatics and mechanics, formalizing the inclined plane as a key simple machine and inspiring quantitative analyses during the early modern period.^[27] By the late 17th century, during the Enlightenment, Isaac Newton's Philosophiæ Naturalis Principia Mathematica (1687) incorporated simple machines into his laws of motion, using examples like levers and pulleys to illustrate force composition and equilibrium in statics and dynamics.^[28] In the Scholium following the definitions, Newton analyzed how machines such as the wheel and inclined plane exemplify the third law of motion through action-reaction pairs, integrating them into a unified framework of classical mechanics that emphasized universal principles over empirical trial.^[29] This synthesis elevated simple machines from practical tools to fundamental illustrations of physical laws, influencing Enlightenment scholars in fields like engineering and philosophy.^[30] In the 19th century, Franz Reuleaux formalized a kinematic theory of machinery in his 1875 book The Kinematics of Machinery, classifying mechanisms through basic kinematic pairs—such as turning pairs and sliding pairs—and chains as the building blocks of all mechanisms, incorporating the classical six simple machines (lever, wheel and axle, pulley, inclined plane, wedge, and screw) as key examples.^[31] Reuleaux's approach treated these elements as pairs of relative motion (sliding or turning), providing a theoretical structure for analyzing machine design through topology rather than isolated functions.^[3] This work marked a shift toward modern kinematics, enabling engineers to synthesize complex systems rationally during the late Industrial era. The Industrial Revolution amplified the role of simple machines in powering steam engines, where components like pistons (acting as levers), crankshafts (wheel and axles), and screw valves formed the core of machinery that drove factories and transportation.^[32] James Watt's improvements to the steam engine in the 1760s–1780s relied on these elements to convert thermal energy into mechanical work, with inclined planes and wedges facilitating assembly and operation in textile mills and ironworks.^[33] This proliferation of compound machines, built from simple ones, transformed economies by enabling mass production and urbanization, as seen in the widespread adoption of steam-powered looms and locomotives by the mid-19th century.^[34]^[35]

Ideal Mechanical Advantage

Fundamental Principles

Simple machines operate under ideal conditions where energy losses are absent, allowing for the direct application of fundamental physical principles to describe their behavior. These principles apply to the six classical types of simple machines: the lever, wheel and axle, pulley, inclined plane, wedge, and screw.^[5]^[36] The core principle governing ideal simple machines is the conservation of work, which states that the work input equals the work output. Work is defined as the product of force and distance, so W_{\text{in}} = F_{\text{in}} \times d_{\text{in}} = F_{\text{out}} \times d_{\text{out}} = W_{\text{out}}, where F_{\text{in}} and d_{\text{in}} are the input force and distance, and F_{\text{out}} and d_{\text{out}} are the output force and distance. This equality holds because, in the ideal case, no energy is dissipated, ensuring that the machine merely transforms the input effort into output without creating or destroying mechanical energy.^[5]^[36] Mechanical advantage (MA) quantifies the force amplification provided by a simple machine and is given by the ratio of output force to input force: \text{MA} = \frac{F_{\text{out}}}{F_{\text{in}}}. From the work conservation principle, this simplifies to \text{MA} = \frac{d_{\text{in}}}{d_{\text{out}}}, indicating that any increase in output force corresponds to a proportional increase in the distance over which the input force acts. The velocity ratio (VR), defined as the ratio of input distance to output distance \text{VR} = \frac{d_{\text{in}}}{d_{\text{out}}}, equals the ideal mechanical advantage under these conditions, reflecting the trade-off between force and speed in the machine's operation.^[5] These principles rely on key assumptions: the machines are frictionless, meaning no dissipative forces oppose motion; the components are rigid bodies that do not deform under load; and there are no other energy losses, such as from heat or inelastic processes. These idealizations allow for straightforward analysis but represent simplifications of real-world behavior.^[5]^[36]

Calculations for Inclined Plane, Wedge, and Screw

The ideal mechanical advantage (IMA) of an inclined plane is derived from the principle that, in the absence of friction, the work input equals the work output. To lift a load of weight W through a vertical height h, the effort force F acts parallel to the plane over its length L. Thus, F \cdot L = W \cdot h, yielding \frac{W}{F} = \frac{L}{h}. Since L = \frac{h}{\sin \theta} where \theta is the angle of inclination, the IMA simplifies to \frac{1}{\sin \theta}.^[1] Alternatively, using force balance, the component of the weight parallel to the plane is W \sin \theta, which equals the effort force required for constant velocity motion. Therefore, IMA = \frac{W}{F} = \frac{W}{W \sin \theta} = \frac{1}{\sin \theta}. This trigonometric derivation highlights how a smaller angle \theta increases the advantage by lengthening the plane.^[37] The wedge functions as a movable double inclined plane, where the effort force drives the wedge to separate or lift loads on both sides. For a symmetric double wedge with apex angle \alpha, each face forms an incline angle of approximately \alpha/2 with the centerline. The IMA is derived by considering the geometry: when the wedge advances a distance s along its length, the lateral displacement on each side is s \tan(\alpha/2), leading to a total separation of $2s \tan(\alpha/2). From energy conservation, input work F \cdot s equals output work over the separation distance, so IMA = \frac{s}{2s \tan(\alpha/2)} = \frac{1}{2 \tan(\alpha/2)}.^[37] Torque equilibrium provides another view: the effort torque balances the resistive torque from the normal forces on the sloped faces, resolved via trigonometry to yield the same angular dependence. This derivation treats the wedge as compounded inclines, where the effective length-to-height ratio doubles due to bilateral action.^[37] The screw achieves mechanical advantage by wrapping an inclined plane around a cylinder, forming a helix. The pitch p represents the vertical advance per revolution (analogous to height h), while the circumference $2\pi r (with r as the radius to the effort point) is the effective length L of the unwrapped plane. Thus, IMA = \frac{2\pi r}{p}, derived from energy conservation where rotational work \tau \cdot 2\pi = F \cdot p (with \tau = F r) simplifies to the linear ratio.^[37] In force terms, the axial effort force balances the component along the helical thread, akin to W \sin \phi where \phi is the lead angle (\tan \phi = p / (2\pi r)), yielding IMA = \frac{1}{\sin \phi} \approx \frac{2\pi r}{p} for small \phi. This helical geometry converts rotary motion to linear with high advantage for fine pitches.^[1]

Real-World Considerations

Friction and Losses

In real simple machines, friction arises from interactions between contacting surfaces, leading to energy losses that reduce the actual mechanical advantage compared to the ideal case.^[38] These losses manifest as heat generated during motion, dissipating useful work and causing deviations from theoretical performance. Historical quantification of such effects began with Charles-Augustin de Coulomb's experiments in the late 18th century, where he investigated friction in machinery using inclined planes and pulleys to establish foundational laws.^[39] Coulomb's 1781 treatise, Théorie des Machines Simples, demonstrated that frictional resistance is proportional to the normal force and independent of contact area or velocity, providing the basis for modern understanding of losses in devices like levers and screws.^[40] Friction in simple machines occurs in two primary regimes: static and kinetic. Static friction acts between stationary surfaces, preventing initial motion until the applied force exceeds the maximum static frictional force, which is given by f_s \leq \mu_s N, where \mu_s is the coefficient of static friction and N is the normal force.^[38] In a lever, for instance, static friction at the pivot (fulcrum) resists rotation until the effort overcomes it, ensuring stability but requiring additional force to initiate movement. Kinetic friction, conversely, opposes motion once it begins, with magnitude f_k = \mu_k N, where \mu_k is typically lower than \mu_s. This type dominates during operation, such as sliding along an inclined plane or rotation at a lever's pivot. Within kinetic friction, sliding friction occurs when surfaces move parallel to each other, as in a block on an inclined plane, while rolling friction applies to wheeled axles or pulleys, where deformation at the contact point generates lower resistance than sliding.^[41] The energy dissipated by friction is the work done against the frictional force, primarily converted to thermal energy. For sliding or kinetic cases, this lost work is calculated as W_{\text{lost}} = \mu_k N d, where d is the distance of sliding, representing the irreversible energy loss per cycle of machine operation.^[42] In a lever's pivot, rotational friction similarly dissipates energy proportional to the torque and angular displacement, contributing to overall inefficiency. Several factors influence the magnitude of these frictional forces and losses: material properties, such as the hardness and elasticity of contacting surfaces, determine adhesion and deformation; lubrication introduces a fluid layer that separates surfaces, reducing direct contact and coefficients by up to orders of magnitude; and surface roughness amplifies friction through interlocking asperities, with smoother finishes lowering \mu values.^[43] For example, unlubricated metal-on-metal contacts in a screw exhibit high roughness-induced friction, while oiled bearings in pulleys minimize losses.^[44]

Efficiency and Power Transmission

Efficiency in simple machines quantifies how effectively the device converts input work or power into useful output, accounting for losses primarily due to friction. It is defined as the ratio of useful output power to input power, expressed as a percentage: \eta = \frac{P_\text{out}}{P_\text{in}} \times 100\%. ^[45] Since power is the rate of work done, this is equivalent to the ratio of output work to input work over the same time interval. In terms of mechanical advantage, efficiency relates the actual mechanical advantage (MA_real, which incorporates friction) to the velocity ratio (VR, the ideal mechanical advantage assuming no losses): \eta = \frac{\text{MA}_\text{real}}{\text{VR}} \times 100\%. ^[46] Friction, such as sliding or rolling types, directly reduces MA_real by requiring additional input force, thereby lowering overall efficiency. Power transmission in simple machines occurs through the fundamental relation P = F \times v, where P is power, F is force, and v is the velocity of the point of application. ^[47] In ideal frictionless conditions, input power equals output power, conserving energy as the machine trades force for velocity (or vice versa) according to the velocity ratio. However, real-world losses from friction, heat, and deformation dissipate energy, reducing transmitted power; for instance, only a fraction of input power may reach the load due to these inefficiencies. This conservation principle, adjusted for losses, underscores why no simple machine exceeds 100% efficiency, with practical values depending on design, lubrication, and operating conditions. Efficiency is measured experimentally using devices like dynamometers, which quantify torque and rotational speed to compute mechanical power output (P = \tau \omega, where \tau is torque and \omega is angular velocity), compared against input power from applied forces or electrical sources. For example, well-lubricated pulley systems can achieve efficiencies of 80-95%, levers often exceed 90% with low-friction pivots, while screw jacks typically range from 20-40% due to higher sliding friction.^[1]^[48] These measurements highlight the impact of maintenance, such as lubrication, on performance.

Compound and Complex Machines

Formation of Compound Machines

A compound machine is formed by combining two or more simple machines, typically in series, to achieve greater mechanical advantage and perform more complex tasks than a single simple machine could accomplish alone. In a series combination, the output force from one simple machine becomes the input force for the next, allowing the advantages to multiply sequentially; a classic example is the block and tackle pulley system, where multiple pulleys are arranged to lift heavy loads by distributing the effort over greater distances. Parallel combinations, where multiple simple machines operate simultaneously to share the load, enable handling larger total loads by distributing the effort, but the overall mechanical advantage remains equivalent to that of the individual machines.^[49]^[50]^[51] The total mechanical advantage of a compound machine depends on the configuration of its components. For machines connected in series, the overall mechanical advantage is the product of the individual mechanical advantages of each simple machine, enabling significant force amplification as seen in systems like gear trains derived from wheels and axles among the six classical simple machines. In parallel arrangements, such as multiple levers supporting a common load, the mechanical advantage does not sum but stays the same as the individual components, as the load is shared equally while the total input force is distributed accordingly, which can enhance stability under load. This multiplication in series or load-sharing in parallel underscores why compound machines are more effective for practical applications requiring substantial force or speed adjustments.^[52] Effective design of compound machines requires careful consideration of velocity ratios—the ratio of input distance or speed to output distance or speed for each component—to ensure compatibility and prevent bottlenecks. If the velocity ratio of one simple machine does not align with the next, it can create inefficiencies, such as mismatched speeds that halt motion or increase wear; thus, engineers match these ratios to maintain smooth power transmission throughout the system. For instance, in pulley-based compounds, aligning the rope travel distances across blocks avoids slack or overload in any segment.^[53] A notable early example of a compound machine is James Watt's steam engine, developed in the 1780s, which integrated levers and beams to link the piston's linear motion to a pump's reciprocating action, demonstrating how combining simple machines could revolutionize power generation. This design leveraged the lever principle of the pivoted beam to amplify force while minimizing energy loss in early industrial applications.^[54]

Practical Examples and Applications

Compound machines, formed by combining multiple simple machines, enable more efficient performance of tasks by multiplying mechanical advantages. Scissors exemplify a compound lever system, where two first-class levers pivot around a fulcrum to amplify cutting force applied by the hand.^[55] Similarly, a bicycle integrates wheels and axles with a gear system, allowing riders to convert pedaling effort into varied speeds and torques through interlocking toothed wheels that function as compound wheel-and-axle assemblies. A hydraulic jack, often incorporating a screw mechanism driven by a lever handle, combines these elements to lift heavy loads like vehicles by translating rotational input into linear elevation with hydraulic assistance.^[50] In construction, cranes utilize compound pulley systems, where multiple fixed and movable pulleys in a block-and-tackle arrangement reduce the force needed to hoist substantial weights, such as building materials or equipment.^[50] In transportation, automotive differentials employ compound gear trains—essentially interconnected wheels and axles—to distribute torque to the wheels, enabling smooth turning by allowing differential speeds without loss of traction.^[56] These compound configurations provide benefits such as enhanced force multiplication for heavy lifting or increased speed for motion, as seen in the bicycle's gear shifts that trade torque for velocity.^[57] However, they introduce trade-offs in added complexity, including higher manufacturing costs and potential points of mechanical failure compared to single simple machines.^[58] In 21st-century applications, robotic arms in manufacturing and exploration draw on these principles, combining levers, pulleys, and screws to achieve precise movements for tasks like assembly or sample collection in underwater environments.^[59]

Self-Locking Machines

Definition and Criteria

A self-locking machine is defined as a mechanical system capable of maintaining a load in position without the application of continuous input force, relying on friction to resist reverse motion once the initial force is removed.^[60] This property is particularly relevant in simple machines such as inclined planes, wedges, and screws, where the static friction prevents the load from causing unintended movement in the opposite direction.^[61] In essence, self-locking ensures stability under load without additional braking mechanisms, making it a key feature for holding applications. The primary criterion for self-locking in these machines is that the friction angle φ, defined by tan φ = μ (where μ is the coefficient of static friction), must exceed the machine's effective geometric angle.^[62] For an inclined plane, self-locking occurs when μ > tan θ, with θ being the incline angle, meaning the load remains stationary without sliding down.^[63] Similarly, for wedges and screws, the condition involves the friction angle surpassing the wedge angle or the lead angle λ of the screw thread, respectively, ensuring friction overcomes the component of force tending to reverse the motion.^[64] This threshold prevents back-driving and is determined by material properties and geometry. Self-locking mechanisms can be classified as permanent or conditional based on their dependence on operating conditions. Permanent self-locking is inherent to the design, such as in certain square-thread screws where the geometry and friction ensure the condition holds consistently across typical loads.^[62] In contrast, conditional self-locking varies with load magnitude; for example, a screw may lock under lower loads but unwind if the load increases sufficiently to alter the effective friction dynamics.^[62] This distinction is crucial for selecting machines in load-bearing applications. A notable application of self-locking is in devices like vices and screw jacks, where it prevents slippage and maintains clamping or lifting positions under sustained loads without external power.

Mathematical Proof

The self-locking condition for a screw is derived from the equilibrium of forces during attempted reversal of motion. Consider a screw with pitch p, mean radius r, axial load W, and coefficient of friction \mu. The lead angle is \alpha = \tan^{-1}(p / 2\pi r). The condition for no back-driving is that the friction prevents reversal, yielding \mu > \tan \alpha. This is obtained from the force balance on the developed thread, analogous to the inclined plane, where the tangential force required to reverse exceeds the friction capacity otherwise.^[62] A general proof for self-locking in simple machines uses the principle of virtual work, which states that for a system in equilibrium, the total virtual work \delta W for any infinitesimal virtual displacement is zero. For stability against reversal, consider a virtual displacement in the reverse direction: the virtual work of the input effort must be at least equal to the virtual work dissipated by friction, \delta W_{\text{input}} \geq \delta W_{\text{friction}}. If the load's potential energy change \delta U = -W \delta h (negative for lowering) is insufficient to overcome friction losses \delta E_f > 0, then -W \delta h + \delta E_f > 0, implying positive input work is required to reverse. This occurs when the machine efficiency \eta = W h / (W h + E_f) < 50\%, as the limiting reversible case is \eta = 50\% where W h = E_f.^[65] For the inclined plane, the self-locking condition is obtained from static equilibrium under gravity. A block of mass m on an incline of angle \theta experiences gravitational component mg \sin \theta down the plane and normal force N = mg \cos \theta. At the verge of sliding, the friction force f = \mu N = mg \sin \theta, so the minimum coefficient to prevent sliding is \mu = \tan \theta. Thus, self-locking requires \mu > \tan \theta.^[63] These proofs rely on idealized assumptions, such as two-dimensional models with constant friction coefficient, negligible thread deformation, and no slip under load. In reality, variations with load magnitude, surface wear, and lubrication can alter the effective \mu, potentially invalidating the conditions for high loads or dynamic scenarios.^[62]

Modern Theoretical Frameworks

Kinematic Chains and Pairs

In modern theoretical frameworks for simple machines, kinematic pairs serve as the fundamental building blocks, representing the constrained connections between rigid elements that enable controlled relative motion. A kinematic pair is defined as an association between two physical components that imposes specific constraints on their movement, allowing only predetermined degrees of freedom while restricting others. This concept was formalized by Franz Reuleaux in his foundational 1875 treatise, where he emphasized pairs as the elemental units for analyzing machine kinematics. Common examples include the revolute pair, which permits pure rotation about a fixed axis, and the prismatic pair, which constrains motion to linear translation along a single direction. These pairs are classified as lower pairs when the contacting surfaces exhibit relative motion through area or line contact, ensuring durability and precise constraint in practical applications.^[66]^[31]^[67] Kinematic chains extend this idea by linking multiple pairs into assemblies of rigid links, forming the structural basis for more complex mechanisms derived from simple machines. Chains are categorized as open or closed based on their topology: an open kinematic chain, such as a serial robot arm, consists of links connected end-to-end without forming loops, allowing the terminal link to move freely in space with multiple degrees of freedom. In contrast, a closed kinematic chain, exemplified by the four-bar linkage, creates a loop where the end of the chain connects back to the starting point, introducing redundancy that constrains motion and often results in periodic or oscillatory behavior. The mobility of these chains, or their effective degrees of freedom (DOF), is quantified using Gruebler's equation for planar mechanisms: DOF = 3(n - 1) - 2j, where n is the number of links (including the fixed frame) and j is the number of binary joints (each providing one constraint). For spatial chains, the general form expands to DOF = 6(n - 1) - Σc_i, where Σc_i sums the constraints from all pairs; this equation, attributed to H. Kutzbach in 1929, extending Grübler's planar criterion from 1917, accounts for the six possible motions (three translations and three rotations) per link in three-dimensional space, adjusted by the total constraints imposed. These formulations enable engineers to predict and design the controlled motions essential to simple machine principles.^[31]^[68]^[69] Simple machines themselves embody these kinematic concepts at their core, illustrating how basic pairs translate into practical force and motion manipulation. For instance, a lever operates through a revolute or planar pair at its fulcrum, where the pivot constrains the beam to rotational motion in a plane, amplifying input force via geometric leverage while limiting translation. Similarly, the screw functions as a helical pair, coupling rotational input with axial translation through threaded surfaces, which constrains the relative motion to a helical path and converts torque into linear advancement, as seen in jacks or vises. These relations highlight how simple machines are not isolated devices but manifestations of paired constraints that underpin broader mechanical systems.^[70]^[71]^[72] Reuleaux's 19th-century innovations provided the theoretical groundwork for these elements, establishing kinematics as a rigorous discipline separate from dynamics and emphasizing the synthesis of machines from pairs and chains. His framework, which treated mechanisms as assemblages of constrained motions, was significantly expanded in the 20th century through applications in robotics, where open and closed chains form the basis for manipulator design, enabling precise end-effector positioning in serial arms or parallel platforms. This evolution addressed limitations in early models by incorporating multi-loop structures and computational analysis, transforming Reuleaux's principles into tools for modern automation.^[66]^[31]^[73]

Classification and Synthesis Methods

Mechanisms derived from simple machines are classified by their primary function into three main categories: function generation, path generation, and motion generation. Function generation mechanisms correlate an input motion, such as rotation or translation, with a specified output motion to approximate a desired relationship between them.^[74] Path generation mechanisms guide a point on a coupler link to follow a prescribed trajectory, often used in applications like drawing irregular curves.^[75] Motion generation mechanisms, also known as rigid-body guidance, position and orient an entire body to match specified poses, ensuring both location and attitude are controlled.^[76] Mechanisms are further classified by complexity into simple, compound, and general mechanisms. Simple mechanisms consist of basic elements like levers, pulleys, or inclined planes with minimal moving parts to alter force or direction.^[77] Compound mechanisms combine multiple simple machines to achieve more intricate tasks, such as a bicycle integrating wheels, gears, and levers.^[78] General mechanisms extend this to kinematic assemblies with controlled degrees of freedom, distinguishing them from unconstrained structures.^[79] Synthesis methods for mechanisms from simple components involve type synthesis and dimensional synthesis. Type synthesis selects appropriate kinematic pairs—such as revolute or prismatic joints—and link configurations to realize the desired motion type from a given number of elements.^[80] Dimensional synthesis then determines the specific lengths, angles, and proportions of links to satisfy kinematic requirements, often optimizing for precision points where the mechanism matches the target exactly.^[81] A foundational tool in synthesis is Burmester theory, which provides geometric constructions for four-bar linkages to achieve path or motion generation through up to five precision points by identifying circle-point and center-point curves.^[82] Developed in the late 19th century, it enables analytical solutions for planar four-bar designs, influencing modern linkage optimization.^[83] Computer-aided design (CAD) tools have integrated mechanism synthesis since the 1980s, evolving from early graphical methods to comprehensive software for simulation and optimization of linkages.^[84] Programs like LINCAGES, developed in the 1970s and refined through the 1980s, automated graphical synthesis for path and function generation, reducing manual iteration.^[3] By the mid-1980s, CAD systems supported interactive design of multi-body mechanisms, incorporating kinematic analysis directly into engineering workflows.^[85] Post-2020 advancements incorporate AI-driven synthesis to automate and innovate mechanism design, addressing limitations in traditional enumeration. Generative models, such as those using cross-domain learning on large datasets of linkages, synthesize novel kinematic structures for specified trajectories by jointly optimizing topology and motion.^[86] Sequence learning approaches treat mechanism assembly as a predictive task, enabling automation of type and dimensional synthesis for complex functions with reduced human input.^[87] These methods leverage datasets like LINKS, containing millions of planar linkages, to train models that generate diverse, high-performance designs beyond classical constraints.^[88]

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• From Archimedes through the Renaissance, mechanisms were considered to be built from combinations of the six simple machines. German mechanical engineer ...
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Machines in Motion - Galileo's World - The University of Oklahoma
Archimedes (d. 212 B.C.) developed the law of the lever with his Treatise on the Balance. He contributed to arithmetic by devising methods for expressing ...
[5]
12.04.02: Simply Amazing - Yale-New Haven Teachers Institute
There are six simple machines, most of which have few, if any, working parts. They are the inclined plane, wedge, screw, lever, wheel and axle, and pulley. When ...
[6]
9.5 Simple Machines – College Physics
Simple machines are devices that can be used to multiply or augment a force that we apply – often at the expense of a distance through which we apply the force.
[7]
Engineering: Simple Machines - Lesson - TeachEngineering
Aug 16, 2024 · Simple machines are devices with no, or very few, moving parts that make work easier. ... simple machine is accomplishing. But in the ...Introduction/motivation · Lesson Background And... · Vocabulary/definitions
[8]
[PDF] Archimedes, the Center of Gravity, and the First of Mechanics:
The lever is one of the simple machines studied in ancient Greece. The other simple machines were the windlass, the pulley, the wedge and the screw. The.
[9]
6 Simple Machines: Making Work Easier | Live Science
Feb 1, 2022 · The most notable of these are known as the "six simple machines": the wheel and axle, the lever, the inclined plane, the pulley, the screw, and the wedge.Missing: credible | Show results with:credible
[10]
Simple Machines – Introductory Physics for the Health and Life ...
Simple machines are devices that multiply or redirect force, often reducing the required input force by increasing the distance through which the force must be ...
[11]
History - Archimedes - BBC
In mechanics he defined the principle of the lever and is credited with inventing the compound pulley and the hydraulic screw for raising water from a lower to ...
[12]
Simple Machines - TeachEngineering
Students apply the mechanical advantages and problem-solving capabilities of six types of simple machines (wedge, wheel and axle, lever, inclined plane, screw, ...
[13]
Pulley - New World Encyclopedia
It is believed however that Archimedes developed the first documented block and tackle pulley system, as recorded by Plutarch. Plutarch reported that Archimedes ...History · Types of Pulleys · Theory of Operation · Uses of Pulleys
[14]
Basic Mechanics
Simple machines include inclined planes, levers, wheel and axle, pulleys, and screws. ... inclined planes are put back to back to form a wedge. In a wedge ...
[15]
[PDF] The Wonderful World of Simple Machines
This program offers a hands-on demonstration of the six types of simple machines: the lever, the inclined plane, the wedge, the screw, the wheel.
[16]
construction of the top of the egyptian pyramids: an experimental test ...
The study experimentally tests a levering device to lift stone blocks on Egyptian pyramids. Levering devices are proposed to complement ramps, addressing ...
[17]
How were the Pyramids of Giza built? - National Geographic
Aug 20, 2025 · It's generally believed that the Egyptians moved massive stone blocks to the heights along large ramps, greased by water or wet clay, using a ...
[18]
The Balance, the Lever and the Aristotelian Origins of Mechanics
We focus on the special role of the balance with unequal arms in the early development of mechanics, on the interaction of various forms of explanatory practice ...
[19]
Archimedes' Law of the Lever
This is the statement of the Law of the Lever that Archimedes gives in Propositions 6 and 7 of Book I of his work entitled On the Equilibrium of Planes.
[20]
Contributions of Archimedes on mechanics and design of mechanisms
Aug 7, 2025 · In this paper his contribution to mechanics is discussed as related to his results in designing and successfully operating mechanisms.
[21]
Water lifting - Roman Aqueducts
One of the oldest known tools used to raise water from a well, river or lake is the shaduf. This simple mechanism, used since pharaonic and classical times.
[22]
Siege of Syracuse (Polybius)
But Archimedes had constructed artillery which could cover a whole variety of ranges, so that while the attacking ships were still at a distance he scored so ...
[23]
Leonardo da Vinci: Mastermind Exhibition
Based on the basic screw, da Vinci designed novel, previously unknown types of screws with a variety of useful applications. Read more · Minimizing Friction.
[24]
Origins: Da Vinci's Machines
Apr 5, 2012 · In this article, we look at Leonardo Da Vinci's sketches of proto-machine tools, including lathes and screw machines, which he conceived hundreds of years ...
[25]
Machines and Mechanisms Designed by Leonardo
Machines and Mechanisms Designed by Leonardo. Discover some of Leonardo da Vinci's mechanical inventions, recorded in his Codex Madrid I.
[26]
Scientist of the Day - Simon Stevin, Dutch Mathematician
Jun 7, 2022 · Simon Stevin, a Dutch mathematician, physicist, and engineer, was born in Bruges in 1548 and died in 1620; we have no exact birth or death dates.Missing: Waterwights | Show results with:Waterwights
[27]
[PDF] the enigma of the inclined plane from heron to galileo - HAL-SHS
The law of the inclined plane states that the ratio between a weight and the force needed to balance this weight on a given inclined plane is equal to the ...
[28]
From decimal fractions to sand yachts – the unbelievably fertile mind ...
Apr 11, 2018 · His demonstration shows that the effective component of gravity is inversely proportional to the length of the inclined plane. What Stevin is in ...
[29]
The Mechanical Philosophy and Newton's Mechanical Force
Jan 1, 2022 · There is little indication in the Principia's most detailed discussion of simple machines that Newton means to redefine mechanics or reconfigure ...
[30]
Newton's Philosophiae Naturalis Principia Mathematica
Dec 20, 2007 · Law 3: The relative motions of bodies enclosed in a given space are the same whether that space is at rest or moves perpetually and uniformly in ...Missing: machines | Show results with:machines
[31]
[PDF] Newton's Principia : the mathematical principles of natural philosophy
NATURAL PHILOSOPHY,. BY SIR ISAAC NEWTON;. TRANSLATED INTO ENGLISH BY ANDREW MOTTE. TO WHICH IS ADDKTV.
[32]
[PDF] Franz Reuleaux and the History of Kinematics and Theory of Machines
Feb 2, 2002 · This review surveys late 19th century kinematics and the theory of machines as seen through the contributions of the German engineering ...
[33]
Steam engine | Definition, History, Impact, & Facts - Britannica
Sep 29, 2025 · In a steam engine, hot steam, usually supplied by a boiler, expands under pressure, and part of the heat energy is converted into work.
[34]
Steam Engines and the Industrial Revolution - ThoughtCo
May 1, 2025 · Steam engines powered factories, made mining easier, and boosted iron production. They also powered transport, and were a key advance of the ...
[35]
The Steam Engine in the British Industrial Revolution
Feb 8, 2023 · Machinery allowed for larger factories, which called for a greater number of workers, and so more people began to live in towns and cities.
[36]
04.04.03: Simple Machines, Engines and the Environment
Let's take a closer look at the steam engine. We know that it uses some of the simple machines such as the wheel and axle as part of the total system (train).
[37]
9.5 Simple Machines - College Physics 2e | OpenStax
Jul 13, 2022 · The ratio of output to input force magnitudes for any simple machine is called its mechanical advantage (MA). MA = F o F i MA = F o F i.Missing: velocity | Show results with:velocity
[38]
Simple Machines - HyperPhysics Concepts
The incline is one of the so-called "simple machines" from which many more complex machines are derived. By pushing an object up a slanted surface, ...
[39]
Simple Machines - Roy Mech
Mechanical Advantage = W /F. The Velocity Ratio of a machine is the ratio of ... Velocity ratio (Double Wedge) = 2 /Tan(θ/2). Lever. Velocity Ratio (First ...
[40]
6.2 Friction - University Physics Volume 1 | OpenStax
Sep 19, 2016 · The equations given for static and kinetic friction are empirical laws that describe the behavior of the forces of friction. While these ...Missing: machines pivot levers Coulomb 1785 experiments
[41]
The Contribution of Coulomb to Applied Mechanics - ResearchGate
Coulomb's major work on friction forces appeared in his “Théorie des Machines Simples” in [3] which won him the Grand Prix from the “Académie des Sciences” ...
[42]
Charles Augustin Coulomb (1736 - 1806) - Biography - MacTutor
His studies into friction in Rochefort led to Coulomb's major work on friction Théorie des machines simples T. (Theory of simple machines). which won him the ...
[43]
Types of Friction | CK-12 Foundation
There are four types of friction: static, sliding, rolling, and fluid friction. Static, sliding, and rolling friction occur between solid surfaces.
[44]
Energy dissipated by friction (video) - Khan Academy
May 2, 2017 · no, we assume that friction is given by the equation (coefficient of friction)*normal force. That doesn't vary. also the formula works as long as you use ...Missing: machines N
[45]
Factors Affecting Friction - Lesson - TeachEngineering
Jun 7, 2019 · Their data analyses reveal the importance of weight to normal friction (the friction that occurs as a result of surface roughness) and the ...Missing: lubrication | Show results with:lubrication
[46]
Frictional Dissipation - an overview | ScienceDirect Topics
... ) W = H π a 2 / 2 = H π x 2 tan 2 α. The coefficient of friction due to the ploughing term will therefore be. (3.11) µ d e f = F d e f / W = 2 / π cot α. A ...Missing: formula | Show results with:formula
[47]
[PDF] Physical Science: Tables & Formulas
Efficiency of a Machine = (Useful Work Output ÷ Work Input) x 100. Temperature Conversions. Celsius-Fahrenheit Conversion: Fahrenheit temperature = (1.8 x ...
[48]
[PPT] Energy - CSUN
Ordinary machines are typically complicated combinations of simple machines. ... Another way to calculate efficiency is by the formula: eff = A.M.A.. I.M.A..<|control11|><|separator|>
[49]
PHYS 200 - Law of Conservation of Energy in Higher Dimensions
I'm going to demand that that's the power, force times velocity, is what I call the power. And I'm going to look for dK/dt in two dimensions, of a body ...
[50]
Power Screws - Mechanics Map
To find the self locking angle, we will assume impending motion (relating the friction force to the normal force) and leave the lead angle as an unknown. This ...Missing: overhauling | Show results with:overhauling
[51]
https://rigginglabacademy.com/parallel-hauls-doubled-mechanical-advantage/
[52]
[PDF] Simple Machines | NASA
A compound machine is a machine consisting of two or more simple machines. Some examples of compound machines are clippers, a manual pencil sharpener, a crane, ...Missing: parallel | Show results with:parallel
[53]
[PDF] Machines At Work
Oct 26, 2025 · The mechanical advantage of a compound machine is just the product of the mechanical advantages of the simple machines of which it is ...
[54]
Parallel Hauls Doubled Mechanical Advantage
May 22, 2019 · It's often argued that two 3:1 systems working in parallel produce a 6:1 advantage, or that two 5:1 systems produce a 10:1. On the surface, this ...Missing: sum | Show results with:sum
[55]
Chapter 2. Mechanisms and Simple Machines
Pulleys can also be arranged as a block and tackle. 2.4 Lever. 2.5 Wheel and Axle. 2.6 Wedge. 2.7 Efficiency of Machines. In working out the problems on levers, ...Missing: typical | Show results with:typical
[56]
Steam Engine History - MSU College of Engineering
The steam engine consists of a steam piston/cylinder that moves a large wooden beam to drive the water pump. The engine does not use steam pressure to push up ...Missing: levers | Show results with:levers
[57]
Simple Machines in Engineering | List & Examples - Study.com
Scissors combine two first-class levers to multiply the cutting force applied by your fingers. Even a zipper is a series of interlocking wedges that are ...
[58]
[PDF] Chapter 11 Drive Lines, Differentials, Drive Axles, and Power Train ...
At the same time, the differential transmits engine torque to the drive axles. The drive axles are on a rotational axis that is 90 degrees different than the ...
[59]
[PDF] Basic Machines And How They Work Understanding Basic ...
combination of two or more simple machines working together to perform a complex task. A bicycle, for example, incorporates levers, wheels and axles, and gears.
[60]
Not So Simple - Lesson - TeachEngineering
a combination of two or more simple machines — can affect work more than its individual components.Missing: parallel | Show results with:parallel
[61]
Simple Machines: Robot Building Blocks - NOAA Ocean Exploration
Students design a manipulator arm for a ROV, exploring how simple machines combine to create complex systems and develop a hydraulic actuator.
[62]
Self-Locking: Static vs. Dynamic | GROB GmbH Antriebstechnik
Rating 4.6 (9) Self-locking is generally defined as the ability of a mechanical system to hold a position without requiring an additional force or brake.
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[PDF] 1). To determine the force required to insert and/or remove a wedge.
Self-Locking Wedge: a wedge in which the friction forces large enough to prevent it from being squeezed out. Remarks. 1). Friction forces always opposes the ...
[64]
9.4 Screw Threads - Engineering Statics
In all three cases the thread angle α is the angle between the normal force N and the centerline of the screw, and the friction angle s ϕ s is the angle between ...Missing: criteria θ
[65]
5.4 Inclined Planes - Physics | OpenStax
Mar 26, 2020 · In situations like this, where an object of mass m slides down a slope that makes an angle θ with the horizontal, friction is given by f k = μ k ...
[66]
[PDF] Dry Friction Wedges Screws Belts - Amin Fakhari
Lowering the load when the screw is self-locking. (i.e., 𝜙s > 𝜃 ). If 𝜙s < 𝜃, the screw unwinds under the load (it is necessary to apply a moment to keep it in.
[67]
Law of Machines - NPTEL Archive
A lifting machine is reversible if its efficiency is greater than 50 percent and self locking if its efficiency is less than 50 percent. This can be proved ...Missing: virtual work proof
[68]
Kinematics of machinery, outlines of a theory of machines
Jul 17, 2007 · Kinematics of machinery, outlines of a theory of machines;. by: Reuleaux, F. (Franz), 1829-1905; Kennedy, Alex.
[69]
[PDF] A Few Basic Concepts and Definitions Related to Mechanisms
• Kinematic pair: A kinematic pair refers to the mechanical joint between any pair of ... 17-18: “In the matter of connections between rigid bodies, Reuleaux ...
[70]
Chapter 4. Basic Kinematics of Constrained Rigid Bodies
4.4.1 Gruebler's Equation. The definition of the degrees of freedom of a mechanism is the number of independent relative motions among the rigid bodies. For ...Degrees of Freedom of a Rigid... · Degrees of Freedom of Planar...
[71]
https://clubtechnical.com/kinematic-pair
[72]
[PDF] sme1203 kinematics of machines - unit 1 basics of mechanisms
Kinematic Pair The two links or elements of a machine, when in contact with each other, are said to form a pair. If the relative motion between them is ...
[73]
kinematic Pair | Link | Definition & Types Explained with Images
Mar 24, 2019 · Screw pair (helical pair) Pair of links having turning as well as sliding motion with respect to each other is called screw or helical pair.
[74]
Types of Kinematic Pairs - BrainKart
Nov 16, 2016 · d) Screw pair (Helical Pair): if two mating links have a turning as well as sliding motion between them, they form a screw pair. This is ...
[75]
Franz Reuleaux: Contributions to 19th C. Kinematics and Theory of ...
Aug 8, 2025 · Frank Reuleaux was one of the first engineers to use symbols to represent machines and kinematic pairs [5] . Using a symbol notation he ...
[76]
Chapter 5. Planar Linkages
Linkages have many different functions, which can be classified according on the primary goal of the mechanism: Function generation: the relative motion ...Missing: generators | Show results with:generators
[77]
Synthesis of function-generation mechanisms using variable length ...
In general, kinematic synthesis problems for four-bar mechanisms can be classified into three categories: function generation, motion generation, and path ...
[78]
present): four-bar path and function generation
Oct 13, 2017 · It includes three distinct subcategories: motion generation, path generation and function generation.
[79]
[PDF] Simple and Complex Machines
Simple machines have few parts, like hammers. Complex machines combine several simple machines, such as clocks and bicycles.
[80]
complex machine - engineering 4 kids
Feb 5, 2019 · Complex (or compound) machines combine one or more simple machines together. The result combines properties and functions of the component simple machines.Missing: parallel | Show results with:parallel<|control11|><|separator|>
[81]
Chapter 3. More on Machines and Mechanisms
Mechanisms can be divided into planar mechanisms and spatial mechanisms, according to the relative motion of the rigid bodies.
[82]
Mechanism Synthesis - an overview | ScienceDirect Topics
Type synthesis studies how many mechanisms with different motions can be composed, given a certain number of linkages and kinematic pairs; linkage synthesis ...
[83]
Developments in quantitative dimensional synthesis (1970-present)
Dimensional synthesis is a type of inverse problem in linkage kinematics where the objective is to calculate the linkage dimensions required to achieve ...<|separator|>
[84]
Design of planar four-bar linkage with n specified positions for a ...
The Burmester problem was initially proposed by Ludwig Burmester to analyze the geometric constraints of the linkage directly from the desired movement for a ...
[85]
Burmester Theory for Four-Bar-Band Mechanisms | J. Mech. Des.
A band mechanism includes inter alia a flexible band and a disk profile. Such a mechanism can be used for path generation better than a four-bar linkage.
[86]
Computer Aided Mechanism Synthesis: A Historical Perspective
This paper provides an historical overviewof the developments in computer aided planar linkage synthesis in the time window of 1955 to the present. The origins ...
[87]
Computer-aided design of mechanisms: 1984 and beyond
This special issue on computer-aided design of mechanisms has the objective of representing the state-of-the-art in computer-assisted mechanism analysis and ...<|control11|><|separator|>
[88]
Creative Synthesis of Kinematic Mechanisms - arXiv
Oct 20, 2025 · In this work, we propose a cross domain generative framework that jointly models mechanical structures and their motion trajectories. Trained on ...Missing: post- | Show results with:post-
[89]
Sequence Learning for Kinematic Mechanism Design Automation
Aug 12, 2025 · The overarching goal of mechanism synthesis is to determine the best arrangement of joints and links such that a mechanical system performs a ...
[90]
[PDF] LINKS: A DATASET OF A HUNDRED MILLION PLANAR LINKAGE ...
In this paper, we introduce LINKS, a dataset of 100 million one degree of free- dom planar linkage mechanisms and 1.1 billion coupler curves, which is more.Missing: post- | Show results with:post-